the advantages of ti calculator’s operating techniques in...

The Advantages of Ti calculator’s Operating Techniquesin Math Teaching

JiangPing [email protected] [email protected] Department High-school

northeast yucai school northeast yucai school110179 110179China China

Abstract

The paper describes the teaching effets of educational technology in math teachingand the desired technology operating mode. With the aid of Ti calculator, some effectiveteaching cases are presented, hoping to achieve the effecient combination of educationaltechnology and teaching demands.

1 The educational technology software used in mathe-

matics teaching of high school

School teachers are faced with various educational technologies and applications. Optimisticallyspeaking, these technologies break the traditional teaching and thinking pattern in a revolu-tionary way. But at the same time, too many of them replicate each other in design and effect.User-unfriendliness is another major disadvantage. Therefore, to integrate educational technol-ogy into everyday class, teachers are overloaded with considerable preparations or relearning.So what was meant to facilitate teaching turns out to discourage teaching. To genuinly supportand change the learning mode, educational softwares and applications still have a long way togo.

What are the features of a practical and effective educational technology? The followingprinciples may well cast some light.

1. User-friendly. In everyday teaching, students are expected to master certain educationaltechnology for the convenience of teacher-student communication and knowledge learning,which burdens the students unavoidably. Moreover, assistive technology could not andshould not be the focus of the class. Therefore, educational technology is expected to beuse-friendly and provide best customer experience. If educational technology can be usedas easily as rulers and compasses, teachers will find no difficulty in facilitating learningand fulfilling the teaching demands accordingly.

Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015)

260

2. Consistent with thinking mode. In many cases of teaching, students are required toexplore as much as they can to solve the problems. And this may well lead to unknownor open-ended conclusions, which call for consistent application and operation of thetechnology. However, many educational technology could merely serve as a tool to verifythe conclusion and thus limits the application scale of the technology or even influencestudents’ estimation and distort their way of thinking.

3. Combined with traditional teaching mode. With the changing of technology, teachingmode is changing rapidly as well. Some teaching modes remain for their excellence andeffectiveness. Effective educational technology should be well combined with such teachingmodes, but not concentrate on the technological change and innovation. In educationaldomain, impractical technology and radical educational revolution are what we face onregular basis. Too much emphasis on the technological innovation can do nothing butbackfire.

4. Presenting graphic mathmatical structure. The abstractness and graphicness are the twointerwined features of mathematics. High abstractness gives rise to deep connotation-s, whereas graphicness conveys the external beauty. Effective educational technologypresents the beauty and complexity of mathematics in an intuitional way, which is theadvantage and the mission of technology in itself. And this is what both teachers andstudents look forward to.

2 Application of Ti calculator

The above are the requirements from technology users, and I will explain them in detail bypresenting the effective application of Ti calculator in math teaching, hoping to explicitlyexpress to the technology producers what we teachers like and what we desire.

2.1 From Ellipse to multi-oval

In the textbook, ellipse is defined as the set of points, the sum of whose distances from twofixed points F1 and F2 is a constant 2a(2a > |F1F2|).

In order to verify the definition intuitively, students are able to use Ti calculator by followingthe below steps:

1. Create a graphs document

2. Set three points F1, F2, P ,connect PF1, PF2 and measure the length.

3. Insert text a + b

4. Click “calculate” and click a + b,select the length of PF1 and PF2

5. right click a + b,select property, lock the data.

6. Press “trace” then select “geometry trace” , drag Pand get the locus.


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Figure 1:

The highlight of this case is the function of lock,which cannot be realized by other softwares.Usually, teachers need to draw a ellipse and check backward whether the sum of distance isa constant. But this conflicts with the proper way knowledge is developed. If we concealthe already-drawn ellipse and simulate this course, it would arouse students’ doubt about itsauthenticity, which is not better than physical teaching aids. And this is why I mentions earlierthat the design of the technology should be consistent with thinking mode. In this case wecan easily develop the problem. With slight changes, we can easily draw the conclusion thathyperbolic and lemniscate are a constant.

We can also make some changes in the above case.

1. Press “trace” and “erase geometry trace”

2. Right click the data of a + b and unlock

3. change text a + b into ab

4. lock the data of ab, trace and get the locus

Figure 2:

When students observe teacher’s operation process, they will find no difficulty in understandingits purpose, and trust the authenticity of such operation just like when they see teachers use


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rules and compasses. At the same time, they can pick up the skill promptly and explore bythemselves. We can even change the problem into an open-ended one, introducing the problemof multi-point constant. Thus, a graphic learning process is formed and learning interestsaroused. Combined with traditional ways, the process paves the way for a better subsequentlearning in the future.

One flaw needs to be mentioned here. The set of points cannot be made into a curve.But in many softwares, this can be realized. Therefore, if this function can be developed, themathematical structure would be intuitive and graphic.

2.2 Conic sections cutting

The verifying of conic section definition is a typical teaching case in mathematic class, whichcannot be effectively explained through static graphs or planar figure. This is when Ti calculatorcomes in handy. It can not only verify the conic section definition, but also demonstrate thecross section curve and cross section curve. All these can be realized with the function of zeros.

1. Create a new document, select left right column, and create graphs in both.

2. Activate the right column, click “view”→“3d graphing” to change it into 3d, and get a3d coordinate system.

3. Use the function of “3d graph entry/edit” in coordinate system, and get the input box

z1(x,y)= ,insert (x2 + y2)12 and get a cone.

4. Insert function z2(x,y)= −(x2 + y2) 12 , and get a infinite cone.

5. In the left column, set 3 parameters within the range from negative to positive number.

6. In the right column(3d), graph z3(x,y)= ax + by + c,and get a plane, the plane willtranslate and rotate with the change of a, b, c

7. In the left column ,use“graph entry/edit”, insert f1(x)=zeros(z3(x,y)-z1(x,y),y) to get theintersect of the plane and cone and get a curve in the left column, insert f1(x)=zeros(z3(x,y)-z2(x,y),y), and get the intersect with another cone.

As with the former case, the model is easily built as if we are cutting with a physical cone.And this is just like what we do in physical experiments. The advantages of the simulatedexperiment embodies the values of innovation of educational technology. Such technology isuser-friendly and feasible, consistent with thinking mode. Unfortunately, it is rare to find. Infront-line teaching, what we need is not complicated functions and techniques but practical andrealistic in everyday teaching. But in the reference material of Ti calculator, zeros functionis not included as the major function. Maybe this is because of the differences of perspectivebetween technology developers and users, or maybe because of the lack of interactions betweenboth sides. Personally speaking, the application of Ti in engineering calculation is more maturethan it is in education. Users are provided with various calculating problems and methods in theofficial manual. If more can be offered in education, Ti calculator would enjoy more popularity.


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Figure 3:

2.3 Normal distribution curve

The teaching of statistics in middle school tend to apply conclusion teaching by giving somemodels without the explanation of principles. Therefore students have to follow the same modeto solve such problems. The model of normal distribution is jsut the case. Without the learningof principles, students cannot have a clear picture of normal distribution curve, which can besimulated with binomial distribution.

1. Create a document with one line above and two columns below.

2. Create a graph and geometry document in the lower left column, insert a slide v1 withthe range from 0 to 12, store the value of 1 to step size

3. Create a list & spreadsheet document in the top line, name line A mm, line B nn. Insert

mm:=seqgen(((ncr(’v1,n))/(2^(’v1))),n,u,{0,’v1},{},1)

in the formula of Line A, insert

nn:=seqgen(((n)/(’v1)),n,u,{0,’v1},{},1)

in the formula of Line B (the formula is a bit complicated but with the message of inputbox, the meaning of parameter is fairly obvious, which aims to calculate the combinationnumber Cin(i = 0, 1, · · ·n) and series in(i = 0, 1, · · · , n)

4. Graph(in, Cin

)in the lower right column, create a data and statistics document, add

variable nn to horizontal ordinate, add mm to longitudinal coordinate, and get a graph.connect Data point ,and get the simulation of Probability density curve. Adjust the slideand get the variation of data.


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Figure 4:

Such experiments are of special meaning in the teaching of elementary mathematics. Asmentioned above, such principles are not strictly presented for the reason of staged teachinggoal setting, which only requires the basic understanding. But the consistency of mathematicalthinking mode leads to the urgent need for students to know how it works. Therefore, avivid and graphic explanation is a must, which narrows the distance between principles andits application. Some similar simulated experiments in statistics also require the help the Titeaching software, which is mainly used to present in an intuitive way. If it can be operatedeasily, students will have the chance to redo and improve the experiments. Ultimately, suchproblems will be solved by experiment.

2.4 Fractal

One obvious advantage of Ti calculator is the integration of different functions and demonstrat-ing Barnslay Pteridium in Fractal is one typical example. During this process, the function ofrandom number, piecewise function and iteration will be used. Such functions can be found inthe basic function of Ti calculator, which means Ti is just like a perfect tool kit, convenientand practical. And students can focus on how to solve the problem but not how to use the toolin their exploration.

1. Insert the parameter of Barnslay Pteridium into the sheet of Ti calculator, group P is theprobability of each part.


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Figure 5:

2. Create functions of f1,f2,f3,f4 as segmentation function, tell which part the random num-ber will be. The parameter here are from step 1.

f1

Define f1(x)=

Func

If x

Func

If x>p[1]+p[2]+p[3] Then

1

Else

0

EndIf

EndFunc

3. Create a sheet with groups of A,B,C. store the value of 1 to A, 2 to B, a random numberto C ( insert =rand().) Insert into a[2] the below formula:

=(’ai[1]*a[1]+’bi[1]*b[1]+’ci[1])*f1(c[1])+

(’ai[2]*a[1]+’bi[2]*b[1]+’ci[2])*f2(c[1])+

(’ai[3]*a[1]+’bi[3]*b[1]+’ci[3])*f3(c[1])+

(’ai[4]*a[1]+’bi[4]*b[1]+’ci[4])*f4(c[1]).

(The algorithm of this formula is simple because it is the plus result of the upper fourformulas, which calculate (a[1],b[1]) by the former transformation. And multiply the fourresults with f1(c[1]),f2(c[1]),f3(c[1]),f4(c[1]), one to each. Because there is only one fromf1(c[1]),f2(c[1]),f3(c[1]),f4(c[1]) equals 1 and other three all equal 0, the overall calculationis to put (a[1],b[1]) into random iteration by one of the four possibilities.) And as thesame, insert the formula into b[2]

=(’di[1]*a[1]+’ei[1]*b[1]+’fi[1])*f1(c[1])+

(’di[2]*a[1]+’ei[2]*b[1]+’fi[2])*f2(c[1])+

(’di[3]*a[1]+’ei[3]*b[1]+’fi[3])*f3(c[1])+

(’di[4]*a[1]+’ei[4]*b[1]+’fi[4])*f4(c[1])

4. Click Line 2, pull down and get a series of data.

Figure 6:


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Figure 7:

5. Create a new graph, draw a scatter diagram, and the date are from a,b. and get aBarnslay Pteridium.

This problem is what I’m faced with in my selective class of junior high mathematics. Studentsare introduced to some interesting problems, and are asked to change them into mathematicalproblems, which calls for a perfect tool kit. And this is also a new orientation of education. Thepresent teaching mode emphasizes the completeness of teaching system, presenting knowledgein a gradual and progressive way. But under the new circumstance, teaching can be driven byproblems and questions. Teachers can summarize some questions and students try to figureout the inner connection among different operating stages with the aid of technological tools.This, not calculation, is the purpose of mathematical education:to prepare students with thecapability to solve problems. Take fractal for example, we do not go to deep lecturing. Wejust tell students all the steps, which can be realized with the calculator. When all steps areintegrated, the whole process turns out to be very easy. Thus, students will spend more timeand energy on the problem itself. we used some other softwares before, which could not realizethe function of piecewise function and random number directly. Thus students were distractedfrom the problem and more attention had to be focused on technology. We even had to opena selective class about the software function before coming to study the major problem. Bydoing so, we put the cart before the horse. What a waste of time. By contrast, Ti is a betterchoice.

From the above we can see that even in the excellent cases, flaws and deficiencies can stillbe found. We have to admit that any kind of demonstration technology has its own weaknessinevitably. For example, Ti’s screen is not very exquisite. Keyboard operation is a little bithard to learn. But educational goals cannot be achieved with technology solely, but hopefullywith the discussion above, more light can be cast on the front-line teaching demands, and aneffective integration between teaching and technology can be realized to make more progressat present and achieve better in the future.


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References

[1] Deborah J.Bennett.“Randomness”.Harvard University.1998

[2] Sun Bowen.“Fractal algorithm and program design”.Science Press.2004

[3] Ti Education.teacher’s home.https://education.ti.com/zh-CN/china/teachers/teachers.2015


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ATCMProceedings2015_CombinedInvited.pdf3892015_20771-ToddSolve First – Ask Questions Later: discovering geometry using Symbolic Geometry and CASSaturday Academy2. An Inscribable Circumscribable Pentagon3. Limiting forms of Triangle Defined Circles4. Telescope Aberration5. Solar Cookers5. Cusps on Circle Caustics6. ConclusionReferences

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3892015_20829-VallejoIntroductionThe log function: Entropy and genomicsAnalytic Geometry: Cassegrain antennasPolynomials: Reed–Solomon corrector methodConclusions

3892015_20831-Jingzhong3892015_20849-HoIntroductionDisciplinarity of codingProblem posing and solvingUnderstanding the problemDevising a planCarrying out the planChecking and extending

Framework for computational thinkingProfessional Development for TeachersA course on computing for teachersMini Project

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3892015_20919-Li3892015_20920-Oeyen3892015_20922-Yao3892015_20930-Wei3892015_20931-Jiajia3892015_20934-PingThe educational technology software used in mathematics teaching of high schoolApplication of Ti calculatorFrom Ellipse to multi-ovalConic sections cuttingNormal distribution curveFractal

3892015_20935-PingPlatonic SolidsArchimedean SolidConstruct Through Truncation

Construction Through Edge Cutting and Corner Truncating Catalan SolidStellationConstructing Polyhedron through ExtendingConstruction by Mutual Containing

3892015_20940-Wei3892015_20941-Li3892015_20943-Yubing3892015_20944-Dahan3892015_20947-Sato3892015_20948-NagaiIntroductionBoolean ring of a power set algebraComputation of Boolean Gröbner bases of a finite powerset algebraEfficient BGB softwareCoding in SageMathComputation DataHierarchy of Sudoku puzzlesConclusions and Remarks

3892015_20958-Kitamoto3892015_20974-Fukuda3892015_20975-Fukuda3892015_20988-McAndrewIntroductionThe SurveyDiscussion of the surveyConclusions

3892015_21034-Makishita3892015_21045-Yingprayoon

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